Open Access

An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces

Boundary Value Problems20092009:628916

DOI: 10.1155/2009/628916

Received: 30 January 2009

Accepted: 15 May 2009

Published: 22 June 2009

Abstract

The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness.

1. Introduction

The theory of fractional differential equations has been emerging as an important area of investigation in recent years. Let us mention that this theory has many applications in describing numerous events and problems of the real world. For example, fractional differential equations are often applicable in engineering, physics, chemistry, and biology. See Hilfer[1], Glockle and Nonnenmacher [2], Metzler et al. [3], Podlubny [4], Gaul et al. [5], among others. Fractionaldifferential equations are also often an object of mathematical investigations; see the papers of Agarwal et al. [6], Ahmad and Nieto [7], Ahmad and Otero-Espinar [8], Belarbi et al. [9], Belmekki et al [10], Benchohra et al. [1113], Chang and Nieto [14], Daftardar-Gejji and Bhalekar [15], Figueiredo Camargo et al. [16], and the monographs of Kilbas et al. [17] and Podlubny [4].

Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq1_HTML.gif , and so forth. the same requirements of boundary conditions. Caputo's fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see [18, 19].

In this paper we investigate the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral conditions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ1_HTML.gif
(11)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq2_HTML.gif is the Caputo fractional derivative, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq4_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq5_HTML.gif are given functions satisfying some assumptions that will be specified later, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq6_HTML.gif is a Banach space with norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq7_HTML.gif .

Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics [20] and cellular systems [21].

Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra [22], Benchohra et al. [23, 24], Infante [25], Peciulyte et al. [26], and the references therein.

In our investigation we apply the method associated with the technique of measures of noncompactness and the fixed point theorem of Mönch type. This technique was mainly initiated in the monograph of Bana and Goebel [27] and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni [28], Guo et al. [29], Lakshmikantham and Leela [30], Mönch [31], and Szufla [32].

2. Preliminaries

In this section, we present some definitions and auxiliary results which will be needed in the sequel.

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq8_HTML.gif the Banach space of continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq9_HTML.gif , with the usual supremum norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ2_HTML.gif
(21)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq10_HTML.gif be the Banach space of measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq11_HTML.gif which are Bochner integrable, equipped with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ3_HTML.gif
(22)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq12_HTML.gif be the Banachspace of measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq13_HTML.gif which are bounded, equipped with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ4_HTML.gif
(23)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq14_HTML.gif be the space of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq15_HTML.gif , whose first derivative is absolutely continuous.

Moreover, for a given set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq16_HTML.gif of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq17_HTML.gif let us denote by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ5_HTML.gif
(24)

Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.

Definition 2.1 (see [27]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq18_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq19_HTML.gif the bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq20_HTML.gif . The Kuratowski measure of noncompactness is the map https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq21_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ6_HTML.gif
(25)

Properties

The Kuratowski measure of noncompactness satisfies some properties (for more details see [27]).

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq22_HTML.gif is compact ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq23_HTML.gif is relatively compact).

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq24_HTML.gif .

(c) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq25_HTML.gif .

(d) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq26_HTML.gif

(e) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq27_HTML.gif

(f) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq28_HTML.gif .

Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq30_HTML.gif denote the closure and the convex hull of the bounded set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq31_HTML.gif , respectively.

For completeness we recall the definition of Caputo derivative of fractional order.

Definition 2.2 (see [17]).

The fractional order integral of the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq32_HTML.gif of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq33_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ7_HTML.gif
(26)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq34_HTML.gif is the gamma function. When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq35_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq36_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ8_HTML.gif
(27)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq37_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq38_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq39_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq40_HTML.gif .

Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq41_HTML.gif is the delta function.

Definition 2.3 (see [17]).

For a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq42_HTML.gif given on the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq43_HTML.gif , the Caputo fractional-order derivative of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq44_HTML.gif , of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq45_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ9_HTML.gif
(28)

Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq47_HTML.gif denotes the integer part of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq48_HTML.gif .

Definition 2.4.

A map https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq49_HTML.gif is said to be Carathéodory if

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq50_HTML.gif is measurable for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq51_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq52_HTML.gif is continuous for almost each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq53_HTML.gif

For our purpose we will only need the following fixed point theorem and the important Lemma.

Theorem 2.5 (see [31, 33]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq54_HTML.gif be a bounded, closed and convex subset of a Banach space such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq55_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq56_HTML.gif be a continuous mapping of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq57_HTML.gif into itself. If the implication
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ10_HTML.gif
(29)

holds for every subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq58_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq59_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq60_HTML.gif has a fixed point.

Lemma 2.6 (see [32]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq61_HTML.gif be a bounded, closed, and convex subset of the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq62_HTML.gif , G a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq63_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq64_HTML.gif satisfies the Carathéodory conditions, and there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq65_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq66_HTML.gif and each bounded set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq67_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ11_HTML.gif
(210)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq68_HTML.gif is an equicontinuous subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq69_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ12_HTML.gif
(211)

3. Existence of Solutions

Let us start by defining what we mean by a solution of the problem (1.1).

Definition 3.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq70_HTML.gif is said to be a solution of (1.1) if it satisfies (1.1).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq71_HTML.gif be continuous functions and consider the linear boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ13_HTML.gif
(31)

Lemma 3.2 (see [11]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq72_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq73_HTML.gif be continuous. A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq74_HTML.gif is a solution of the fractional integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ14_HTML.gif
(32)
with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ15_HTML.gif
(33)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ16_HTML.gif
(34)

if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq75_HTML.gif is a solution of the fractional boundary value problem (3.1).

Remark 3.3.

It is clear that the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq76_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq77_HTML.gif , and hence is bounded. Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ17_HTML.gif
(35)

For the forthcoming analysis, we introduce the following assumptions

(H1)The functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq78_HTML.gif satisfy the Carathéodory conditions.

(H2)There exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq79_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ18_HTML.gif
(36)
(H3)For almost each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq80_HTML.gif and each bounded set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq81_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ19_HTML.gif
(37)

Theorem 3.4.

Assume that assumptions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq82_HTML.gif hold. If
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ20_HTML.gif
(38)

then the boundary value problem (1.1) has at least one solution.

Proof.

We transform the problem (1.1) into a fixed point problem by defining an operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq83_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ21_HTML.gif
(39)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ22_HTML.gif
(310)
and the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq84_HTML.gif is given by (3.4). Clearly, the fixed points of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq85_HTML.gif are solution of the problem (1.1). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq86_HTML.gif and consider the set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ23_HTML.gif
(311)

Clearly, the subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq87_HTML.gif is closed, bounded, and convex. We will show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq88_HTML.gif satisfies the assumptions of Theorem 2.5. The proof will be given in three steps.

Step 1.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq89_HTML.gif is continuous.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq90_HTML.gif be a sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq91_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq92_HTML.gif . Then, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq93_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ24_HTML.gif
(312)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq94_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ25_HTML.gif
(313)
By (H2) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ26_HTML.gif
(314)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq96_HTML.gif are Carathéodory functions, the Lebesgue dominated convergence theorem implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ27_HTML.gif
(315)

Step 2.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq97_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq98_HTML.gif into itself.

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq99_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq100_HTML.gif and (3.8) we have for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq101_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ28_HTML.gif
(316)

Step 3.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq102_HTML.gif is bounded and equicontinuous.

By Step 2, it is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq103_HTML.gif is bounded.

For the equicontinuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq104_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq107_HTML.gif . Then

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ29_HTML.gif
(317)

As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq108_HTML.gif , the right-hand side of the above inequality tends to zero.

Now let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq109_HTML.gif be a subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq110_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq111_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq112_HTML.gif is bounded and equicontinuous, and therefore the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq113_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq114_HTML.gif . By (H3), Lemma 2.6, and the properties of the measure https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq115_HTML.gif we have for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq116_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ30_HTML.gif
(318)
This means that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ31_HTML.gif
(319)

By (3.8) it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq117_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq118_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq119_HTML.gif , and then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq120_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq121_HTML.gif . In view of the Ascoli-Arzelà theorem, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq122_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq123_HTML.gif . Applying now Theorem 2.5 we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq124_HTML.gif has a fixed point which is a solution of the problem (1.1).

4. An Example

In this section we give an example to illustrate the usefulness of our main results. Let us consider the following fractional boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ32_HTML.gif
(41)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ33_HTML.gif
(42)

Clearly, conditions (H1),(H2) hold with

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ34_HTML.gif
(43)
From (3.4) the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq125_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ35_HTML.gif
(44)
From (4.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ36_HTML.gif
(45)
A simple computation gives
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ37_HTML.gif
(46)
Condition (3.8) is satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq126_HTML.gif . Indeed
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ38_HTML.gif
(47)

which is satisfied for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq127_HTML.gif . Then by Theorem 3.4 the problem (4.1) has a solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq128_HTML.gif .

Declarations

Acknowledgments

The authors thank the referees for their remarks. The research of A. Cabada has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.

Authors’ Affiliations

(1)
Laboratoire de Mathématiques, Université de Sidi Bel-Abbès
(2)
Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela
(3)
Département de Mathématiques, Université de Boumerdès

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